AbrahamPhysics

Advanced

Finding the position of a classical particle as a function of time is one of the primary activities of classical mechanics. A procedure goes something like the following: First, calculate the net force on a particle either from the forces themselves or from the potential eneregies. Next, solve for the acceleration using Newton's second law and then integrate to find the position function.

In practice, since the result of applying Newton's laws often doesn't give something you can integrate analytically, you numerically integrate Newton's laws directly for the position function. Since, the acceleration is the second time-derivative of the position, Newton's second law can be written as: $$ \vec{F} = m\frac{d^2\vec{r}\, (t)}{dt^2} $$ This gives a second order differential equation for the position function, \(\vec{r}(t)\). This can be numericaly solved to find \(\vec{r}(t)\) using one of many techniques in numerical methods. One would call this simply, "integrating Newton's equations of motion".

Interestingly, despite playing a central role in classical mechanics, the position function doesn't even exist in quantum mechanics! The idea that a particle travels a specific path in space is not a valid concept in quantum mechanics. A particle samples all possible paths between two points, with probabilities governed by the mathematics of quantum mechanics. While that may seem too crazy at this point, just think how cool it will be when you get there. Also, remember, almost everything you learn at this point in your career is only partially true, or true only under ceratin conditions and approximations.